It has long been recognized that broadband laser light has the potential to control parametric instabilities in inertial-confinement-fusion (ICF) plasmas. Here, we use results from laser-plasma-interaction simulations to estimate the bandwidth requirements for mitigating the three predominant classes of instabilities in direct-drive ICF implosions: cross-beam energy transfer (CBET), two-plasmon decay (TPD), and stimulated Raman scattering (SRS). We find that for frequency-tripled, Nd:glass laser light, a bandwidth of 8.5 THz can significantly increase laser absorption by suppressing CBET, while ∼13 THz is needed to mitigate absolute TPD and SRS on an ignition-scale platform. None of the glass lasers used in contemporary ICF experiments, however, possess a bandwidth greater than 1 THz and reaching larger values requires the use of an auxiliary broadening technique such as optical parametric amplification or stimulated-rotational-Raman scattering. An arguably superior approach is the adoption of an argon-fluoride (ArF) laser as an ICF driver. Besides having a broad bandwidth of ∼10 THz, the ArF laser also possesses the shortest wavelength (193 nm) that can scale to the high energy/power required for ICF—a feature that helps to mitigate parametric instabilities even further. We show that these native properties of ArF laser light are sufficient to eliminate nearly all CBET scattering in a direct-drive target and also raise absolute TPD and SRS thresholds well above those for broadband glass lasers. The effective control of parametric instabilities with broad bandwidth is potentially a “game changer” in ICF because it would enable higher laser intensities and ablation pressures in future target designs.

## I. INTRODUCTION

In the direct-drive approach to inertial confinement fusion (ICF), a millimeter-scale spherical capsule containing nuclear fuel is irradiated with many symmetrically distributed beams of intense laser light.^{1} The lasers ablate the outer layer of the capsule and launch strong, inward-propagating shock waves that compress and heat the enclosed fuel to thermonuclear conditions. To achieve high gain, the lasers must couple their energy efficiently to the electrons in the underdense ( $ne<nc=\pi \u2009me\u2009c2/e2\u2009\lambda 02$) plasma corona surrounding the capsule and those electrons must then thermalize their energy *locally* via collisions with ions. Here, *n _{e}* is the local number density of electrons in the plasma,

*n*is the critical density beyond which laser light of wavelength

_{c}*λ*

_{0}cannot propagate,

*m*is the electron mass,

_{e}*c*is the speed of light, and

*e*is the magnitude of the electron charge. A successful ICF implosion therefore requires that the effects of laser-driven parametric instabilities

^{2}—which can spatially redistribute significant fractions of the incident laser energy and reduce the ablation pressure and finely tuned symmetry of the target illumination—be minimized so that collisional (i.e., inverse bremsstrahlung) absorption is the dominant energy-transfer mechanism at fusion-relevant laser intensities ( $\u223c1014$–10

^{16}$W/cm2$).

There are a number of parametric processes that impair the collisional absorption of laser energy in an under-dense plasma. Featuring prominently in direct-drive ICF is the two-plasmon-decay (TPD) instability, which occurs close to $nc/4$ and in which the incident laser decays into two electron plasma waves, and the stimulated Raman scattering (SRS) instability, which occurs at density values near and below $nc/4$ and in which the incident laser excites a scattered electromagnetic wave of lower frequency and an electron plasma wave.^{2,3} The TPD and SRS instabilities are of particular concern in ICF because they can also accelerate electrons to supra-thermal energies (>50 keV), which can then penetrate the high-density fuel layer of the target, raise its adiabat, and degrade fusion yield by reducing compressibility.^{4–6} In current direct-drive experiments, the percentage of the incident laser energy that is converted into “hot” electrons can exceed 1%, with a significant fraction of this energy potentially being deposited in the cold fuel. Consequently, preheat caused by hot electrons presently limits the overlapped vacuum laser intensity in direct-drive implosions on the OMEGA laser to values less than $\u223c1015\u2009\u2009W/cm2$.^{7} Another deleterious laser-plasma interaction in direct-drive ICF is cross-beam energy transfer (CBET),^{8,9} which is a special form of stimulated Brillouin scattering that can occur when two lasers cross each other near a sonic point in the plasma flow. CBET can occur anywhere below the critical density and can lead to the resonant exchange of laser energy from one beam to another via ion acoustic waves. In recent years, CBET has been identified as a critical obstacle to success in direct-drive ICF because of its ability to reduce the energy coupling efficiency by as much as 50% with existing laser drivers.^{10,11} The prevalence of all three of these instabilities increases with the laser intensity *I*, which imposes significant limitations on the pulse shapes that can be used to drive ICF implosions.

CBET, TPD, and SRS are all resonant three-wave instabilities that satisfy the following energy and momentum conservation relations:

*ω*

_{0}denotes the laser angular frequency and $k0$ is the laser wavevector. Similarly, quantities with the subscripts 1 and 2 represent the frequencies and wavevectors of the two daughter waves. A parametric instability can occur as laser light propagates through a plasma and the electric field of the laser accelerates plasma electrons, which in turn emit secondary radiation via Thomson scattering.

^{12}The beating of the incident laser wave with this scattered light wave gives rise to a ponderomotive force that tends to push plasma particles into the troughs of the beat-wave envelope, causing those particles to bunch together and form a longitudinal wave. If the frequency and wavenumber of this wave happens to satisfy the dispersion relation for one of the possible electrostatic modes in an unmagnetized plasma, the three waves become coupled energetically, and a significant fraction of the incident laser energy can be transferred to the two daughter waves. Such a parametric process constitutes an undesirable laser-absorption mechanism in a plasma that competes unfavorably with the preferred method of collisional absorption.

It has been known since the late 1970s that collisional absorption in a plasma can be significantly improved using short laser wavelengths.^{13} Prior to this time, laser-fusion research programs had primarily employed infrared ( $>1\u2009\u2009\mu m$) drivers due to their high efficiency in generating energetic laser pulses. Such long-wavelength lasers, however, are ill-suited for ICF applications due to their poor energy coupling and were subsequently abandoned in favor of ultraviolet lasers with sub-micron wavelengths. The reason that energy coupling improves as the laser wavelength decreases is because shorter wavelength light is able to penetrate to regions of the plasma that have a higher density and a lower temperature, and both of these factors increase the electron-ion collision rate $\nu ei=4\u20092\u2009\pi \u2009nc\u2009Z\u2009e4\u2009\u2009ln\u2009\Lambda e/3\u2009me1/2\u2009(kBTe)3/2$ (where *Z* is the average ionization state of the plasma, $ln\u2009\Lambda e$ is the Coulomb logarithm, *k _{B}* is the Boltzmann constant, and

*T*is the electron temperature).

_{e}^{1}Moreover, since the ponderomotive force goes roughly as $I\lambda 02$,

^{14}shorter-wavelength laser light drives parametric instabilities less strongly, which enables higher collisional absorption for typical direct-drive laser intensities. For these reasons, most ICF experiments today utilize either a frequency-tripled neodymium-glass (Nd:glass) laser or a krypton-fluoride (KrF) excimer laser, which have wavelengths of 351 and 248 nm, respectively.

Another means of improving laser-plasma energy coupling in an ICF target is to reduce the laser's spatial and temporal coherence—both of which help to increase beam uniformity. Typically, the profile of an un-smoothed, high-power laser beam includes many sizable intensity modulations (“hot spots”) that can drive laser-plasma instabilities and prevent efficient energy coupling. Spatial incoherence helps to smooth the large-scale intensity fluctuations and can be implemented by dividing each beam into multiple beamlets with different phases and/or polarization states and then superimposing those beamlets at the focal plane of the target. Further improvements in uniformity can be realized by introducing temporal incoherence in the form of finite laser bandwidth, $\Delta \nu $, which causes the beam speckle pattern to fluctuate in time so that relatively smooth irradiance profiles can be achieved when averaged over an interval that is large compared to the laser coherence time $tc\u22431/\Delta \nu $. In addition to enabling smoother profiles, large bandwidths can also directly mitigate parametric instabilities^{15} by lessening the amount of laser energy driving the resonant decay process described by Eq. (1). This is a consequence of the fact that the energy in a broadband pulse is distributed over a range of frequencies, which, if sufficiently large, can exceed the resonance width of the instability and thus reduce the effective laser power available to excite it.^{16}

Two well-known beam-conditioning techniques that are based on modifying the spatial and temporal coherence of the laser light are smoothing by spectral dispersion (SSD)^{17} and induced spatial incoherence (ISI).^{18} Experiments performed with high-power ICF lasers incorporating these schemes have demonstrated a reduction by several orders-of-magnitude in the level of SRS for ICF-relevant conditions.^{19,20} Much of the observed suppression, though, is understood to be attributable to modifications in the spatial incoherence of the laser, since the bandwidths available in those experiments should not have been large enough to suppress parametric instabilities directly.^{21} Although not entirely understood, the most plausible explanation for this observation is the suppression of the more-slowly growing (and non-resonant) filamentation instability, which can, for example, create localized regions of high laser intensity in the plasma that amplify SRS gain. Another possibility is that the more uniform irradiance reduces turbulent microstructures in the plasma, which leads to an improvement in the laser-plasma coupling. Such explanations are consistent with other experiments^{22} that have examined the effects of laser bandwidth alone and observed little if any improvement in the laser-plasma coupling for $\Delta \nu \u2009\u2272$ 1 THz.

To date, no ICF experiments have been performed with a laser whose bandwidth is broader than a few terahertz. SSD bandwidths on the frequency-tripled Nd:glass lasers at OMEGA and the National Ignition Facility (NIF), for example, are presently limited to about 1 and 0.3 THz, respectively, due to considerations of optics damage and efficient frequency conversion.^{23} The Nike KrF laser at the U.S. Naval Research Laboratory (NRL), which utilizes ISI beam smoothing, has reached significantly larger bandwidths—about 3 THz (at a wavelength of 248 nm) with spectral shaping of the input beam^{24}—although this is still insufficient to directly mitigate resonant parametric instabilities. As we discuss in greater detail below, numerical simulations suggest that affecting many of the predominant parametric instabilities in direct-drive ICF likely requires bandwidths on the order of 10 THz. One of the principal objectives of this paper is to motivate research into the development of broadband laser sources for testing this prediction experimentally.

Possible approaches for broadening Nd:glass laser bandwidths include the application of certain nonlinear optical methods such as parametric amplification^{25} or stimulated rotational Raman scattering (SRRS)^{26} near the output of the laser. Alternatively, significant instability mitigation can be achieved using the argon-fluoride (ArF) excimer laser, which has a native bandwidth of approximately 10 THz.^{27–29} An additional advantage of an ArF driver is its ultra-short, deep-ultraviolet wavelength of 193 nm, which helps to improve laser absorption and suppress parametric instabilities even further. In this paper, we explore the efficacy of ArF laser light for controlling CBET scattering as well as the absolute forms of TPD and SRS. The effective mitigation of these instabilities in direct-drive ICF would facilitate the use of larger ablation pressures to implode more massive, lower aspect-ratio targets that are more stable hydrodynamically and require less precision in their fabrication and illumination uniformity to achieve high yield. This would greatly expand the accessible parameter space in future, direct-drive target designs.

This paper is organized as follows. In Sec. II, we discuss the important distinction between the absolute and convective forms of parametric instabilities and the impact that broad laser bandwidth has on each. Section III describes the governing equations underlying our theoretical model, which are solved numerically using the fluid-based code LPSE (Laser Plasma Simulation Environment).^{30} Section IV presents results of multi-dimensional simulations performed with LPSE to model CBET and absolute TPD and SRS in the context of direct-drive ICF using 351-nm laser light. In Sec. V, we discuss a promising option for achieving the large bandwidths needed with frequency-tripled Nd:glass lasers for parametric instability suppression, namely, the SRRS technique. In the same section, we present additional LPSE simulations of CBET scattering and show that at its native bandwidth of approximately 10 THz, an ArF laser eliminates nearly all CBET-induced absorption losses in an OMEGA-scale ICF target. Utilizing a methodology developed by Follett *et al.*,^{31} we also compare absolute multi-beam TPD and SRS thresholds between frequency-tripled Nd:glass and ArF lasers for typical ICF-plasma conditions and show that the bandwidth of the former class of drivers would need to be enhanced to values in excess of 20 THz to achieve the same degree of instability mitigation achieved natively with the latter. Finally, in Sec. VI, we conclude this paper with a summary of results presented herein.

## II. CONVECTIVE AND ABSOLUTE PARAMETRIC INSTABILITIES

Theoretical studies^{32,33} on the subject of parametric instability suppression predict that in a uniform, quiescent plasma, a laser with a finite angular bandwidth $\Delta \omega =2\pi \u2009\Delta \nu $ can reduce the growth rate of an instability from *γ*_{0} to $\gamma 02/\Delta \omega $, provided that $\Delta \omega >\gamma 0$. (Note that *γ*_{0} is the growth rate associated with a single, monochromatic laser and that the theory implies *γ*_{0} is unaffected when $\Delta \omega <\gamma 0$.) This prediction has been verified in experiments of the parametric decay instability in which broadband microwaves were used to simulate laser–plasma interactions.^{34} It has also been observed in one-dimensional particle-in-cell simulations modeling the SRS instability.^{35,36} Note, however, that the above result does not generally apply to ICF plasmas owing to the presence of significant inhomogeneities (i.e., density, temperature, and velocity gradients) in the areas of the corona where parametric instabilities develop. Moreover, in an inhomogeneous plasma, the efficacy of broadband laser light for suppressing an instability depends inherently on whether that instability grows convectively or absolutely.

For a convective instability in an inhomogeneous plasma, daughter waves experience exponential growth as they begin to propagate, but this growth ceases once the waves are convected out of the region where the resonance conditions for the decay [viz., Eq. (1)] are approximately satisfied.^{3,37} In this case, the quantity of interest is the convective gain, which is defined as the number of *e*-foldings experienced by an amplified seed pulse. Previous work by Guzdar *et al.*^{38} showed that while finite bandwidth can reduce the growth rate of a convective instability, it simultaneously broadens the extent of the resonance region in such a way that the convective amplification is unchanged; see Fig. 1. It is important to note that in deriving this result, Guzdar *et al.* assume that the growth rate does not vary in space and that all frequencies in the power spectrum of the laser are resonant somewhere within the interaction region.

Even though CBET is a convective instability, it can still be suppressed by broadband laser light because this last assumption is not generally valid in direct-drive scenarios.^{39} For CBET, the region of resonance in a direct-drive ICF target is determined by the flow velocity in the coronal plasma, which typically ranges from values close to zero near the critical surface to a few times the ion acoustic speed $cs=[(ZkBTe+3kBTi)/mi]1/2$ at large radial distances. Here, *T _{i}* and

*m*are the temperature and mass, respectively, of the ion species in the plasma. For 351-nm laser light, numerical simulations presented in Sec. IV A indicate that a Gaussian laser bandwidth with a full-width-at-half-maximum of 8.5 THz ( $\u223ck0cs$ for typical ICF conditions) is sufficient to extend the resonance region into low-density areas of the coronal plasma, where the growth rate is small, or to densities above the critical value, where the laser light does not propagate. In this way, many points of resonance can be excluded from the bulk of the plasma volume, thus allowing broad bandwidth to have an overall mitigating impact on CBET.

_{i}^{40,41}This effect has also been demonstrated in “wavelength-detuning” experiments performed on both the OMEGA

^{42}laser and the NIF

^{43}in which beams with multiple

*discrete*frequencies were used to achieve partial CBET suppression and enhanced laser absorption.

Unlike a convective instability, the absolute variety occurs when daughter waves grow exponentially in time at a fixed location in space.^{44–46} This can occur when the pump laser is sufficiently strong to transfer energy to the daughter waves faster than they can be advected out of the resonant region, resulting in absolute growth that is saturated only by nonlinear effects. For quantifying the mitigation of absolute instabilities with bandwidth, the most meaningful quantity is the threshold laser intensity at which the instability develops. Because absolute instabilities occur at a fixed spatial location, the broadening of the resonance region caused by finite laser bandwidth does not offset the reduction in the growth rate. As an example, consider the case of the absolute, SRS backscatter instability (see the discussion below), which can *only* arise at electron densities near $nc/4$. Resonance points for absolute SRS backscatter that are shifted by broadband laser light to different densities will be stable because the instability cannot occur at such density values. Furthermore, since the total energy in a broadband laser pulse is distributed over a range of frequencies, the absolute SRS-backscatter threshold generally increases with bandwidth^{47} because the effective laser power available to drive the resonance state at $nc/4$ is reduced. This behavior is markedly different from that of convectively unstable SRS modes, which can occur at any electron plasma density below the quarter-critical value. In that case, the spreading of resonance points in space with broad laser bandwidth does not mitigate the instability because convective SRS can still occur at those other densities, as demonstrated in Fig. 1. Note that unlike CBET, the SRS and TPD instabilities exhibit both convective and absolute modes in laser-ablated plasmas.

Although both forms of the TPD instability are usually present in direct-drive ICF experiments, it is the absolute variety that poses the greater concern. This is because TPD typically becomes unstable in an absolute sense when the convective gain is still relatively small.^{48} As a result, absolute TPD generally dominates over its convective counterpart and is primarily responsible for the generation of hot electrons in short-scale-length ICF plasmas irradiated with 351-nm light, such as those produced on the OMEGA laser.^{49,50} Previous studies have estimated the thresholds of absolute TPD for a single broadband beam analytically in one spatial dimension^{47} and numerically in two dimensions.^{31} Recently, Follett *et al.*^{39,51} have extended this analysis to three dimensions and calculated thresholds for multiple, overlapped beams from frequency-tripled Nd:glass lasers under realistic ICF–plasma conditions.

For SRS, the convective and absolute forms of the instability are further classified as backscatter or sidescatter modes, depending on the direction of propagation of the daughter electromagnetic wave with respect to the local density gradient. Thus, there are four different categories of the SRS instability that can develop in an ICF plasma: absolute backscatter, absolute sidescatter, convective backscatter, and convective sidescatter. Absolute SRS backscatter, as mentioned previously, only occurs near the quarter-critical surface, where the daughter electromagnetic wave has a group velocity that is close to zero. The other three modalities of SRS can arise at any density in the coronal plasma that lies below the quarter-critical value. Although convective SRS-backscatter gains are usually small for typical direct-drive plasma conditions, the same is generally not true for convective SRS sidescatter,^{52} which can have large gains due to long effective scale-lengths and, along with the two absolute forms of SRS, can generate significant quantities of hot electrons in ICF targets.

While finite laser bandwidth can be used to raise absolute TPD and SRS thresholds, it is generally ineffective at suppressing the convective forms of those instabilities. This is because the amount of bandwidth required to affect resonant decay processes involving high-frequency electron plasma waves (and stretch their resonance regions to, say, very low densities where enhanced Landau damping would inhibit instability growth) is impractically large.^{39} We should point out, however, that certain kinetic effects such as autoresonance^{53} and kinetic inflation^{54}—which, for example, are known to amplify convective SRS gains through the flattening of the electron velocity distribution function near the electron-plasma-wave phase velocity—can be suppressed with broadband lasers.^{55} An analysis of such effects, though, lies beyond the scope of the present investigation, which is constrained by the fluid-based approximations underpinning our numerical model. Consequently, we limit our attention in this study to examining the effects of broad bandwidth on the absolute forms of TPD and SRS only.

## III. THEORETICAL MODEL

In this section, we present a brief overview of the theoretical framework that is used to model CBET, TPD, and SRS in the present investigation. Our approach for analyzing the effects of broad laser bandwidth on these parametric instabilities utilizes the LPSE code, which solves an extended version of the Zakharov equations for describing the coupling in a laser-irradiated plasma between ion and electron perturbations relative to a neutral background.^{30,56–60} LPSE is capable of simultaneously solving up to four, time-enveloped wave equations for the various electrostatic and electromagnetic modes that can exist in an unmagnetized plasma. In general, these modes consist of electron plasma waves, ion-acoustic waves, Raman and Brillouin-scattered light, and the electromagnetic field of the incident laser. For our purposes, though, it is only necessary to consider a limited set of such modes to adequately model each instability. As we shall see, while the individual equations that LPSE solves are linear, the coupling between different modes can result in nonlinear laser–plasma effects.

^{61}The equation for the former is obtained by first writing $E\u0303=Re[E0(x,t)\u2009exp\u2009(\u2212i\omega 0t)]$, where $E\u0303$ denotes the total electric field of the laser and we assume that the envelope $E0$ varies slowly over its period of oscillation. This assumption is valid provided that the bandwidth of the laser is much less than its central frequency, i.e., $\Delta \nu /\nu 0\u226a1$, where $\nu 0=\omega 0/2\pi $. Note that for the Nd:glass and ArF lasers modeled in this investigation, the above inequality is satisfied for bandwidths well into the tens-of-terahertz regime. With this approximation, and also assuming $\nu ei\u226a\omega 0$, a combination of Faraday's and Ampère's laws yields

*δ*denotes a small perturbation. We further assume that the gradients of static quantities are negligible with respect to those of the perturbations—an approximation that is justified since hydrodynamic length scales in direct-drive ICF plasmas greatly exceed the wavelength of an ion acoustic wave. Using these approximations, we find

^{61}

^{,}

^{62,63}which is implemented in LPSE by applying a constant damping factor $\nu a/2$ in

*k*-space. (Factors of two arise here because

*ν*is an energy-damping rate, whereas $\nu \u0302a$ is an amplitude-damping rate.) Note that LPSE does not adopt a paraxial approximation (i.e., spatial enveloping)

_{a}^{64,65}or require periodic boundary conditions in the transverse directions for CBET simulations. Moreover, in addition to solving Eqs. (2)–(4) in the bulk of the plasma volume, LPSE must also model realistic laser sources and eliminate outgoing waves at the edges of the numerical grid. To do this, LPSE utilizes a total-field/scattered-field formulation

^{66–68}in conjunction with a perfectly matched-layer technique

^{69}that allows the laser light to enter and exit through any boundary of the computational domain at arbitrary angles and with minimal numerical reflectivity. This approach enables LPSE to simulate CBET in a variety of complex geometries, including, for example, configurations in which beams are injected from the corners of the numerical grid.

One option for solving Eqs. (2)–(4) with LPSE is to use an explicit, time-split finite-difference method. The plasma fluid variables *δn* and *w* are leap-frogged in time over each other, while the electromagnetic envelope $E0$ is sub-cycled using a smaller time step and a similar leap-frogging algorithm is used to advance its real and imaginary parts.^{61} A second solution strategy is to solve *δn* and *w* with the same numerical technique, but evolve $E0$ using a spectral method. This approach can reduce run times in LPSE by almost an order of magnitude (compared to a finite-difference-based solution with the same level of accuracy) in some cases, mostly because it allows for larger time steps and requires less spatial resolution. Additional details of the numerical algorithms featured in LPSE can be found in Refs. 39 and 70.

^{57}

^{,}

**E**

_{h}is the complex envelope of the electron-plasma-wave field and $\omega p0=(4\pi e2n0/me)1/2$ is the associated envelope frequency at the reference density

*n*

_{0}such that the total rapidly varying field is $E\u0303h=Re[Eh(x,t)\u2009exp\u2009(\u2212i\omega p0t)]$. Note that in Eq. (5), an asterisk denotes a complex conjugate, $vte=(kBTe/me)1/2$ is the electron thermal velocity, and $\gamma L\u2009\xb0$ is a phenomenological operator that accounts for a combination of collisional and Landau damping of electron plasma waves.

^{58}(The collisional part of $\gamma L\u2009\xb0$ is applied in configuration space using the Braginskii coefficient for electron-ion collisional damping,

^{71}while Landau damping is applied in k-space.) Also note that the term

*S*in Eq. (5) represents a time-random-phase Cerenkov noise source that provides the numerical seed in LPSE from which instabilities grow. In this study, the solution to Eq. (5) was calculated using a pseudo-spectral split-step technique based on fast Fourier transforms, which is discussed in detail in Ref. 39. Since nonlinear saturation mechanisms (e.g., pump depletion and coupling to ion-acoustic perturbations) are not required to estimate TPD thresholds, such terms have been omitted from Eq. (5), which is linear in the daughter wave amplitudes. The absence of feedback between the daughter waves and the laser light permits the electric field envelope $E0$ to be constructed on each time step rather than evolved temporally, which greatly reduces the run time of the simulations.

_{E}^{31,39}

Some of the LPSE simulation results presented in this paper include the effects of polarization smoothing, which is implemented in the code by assigning each beam a random polarization direction and then splitting the total beam intensity equally between that state and a second orthogonal component. This approach is possible because LPSE solves the vector wave equation for $E0$ in three spatial dimensions. To model the effects of a distributed phase plate (DPP), each beam is constructed from a superposition of 4th-order-super-Gaussian plane waves with slightly different propagation angles. The basic idea is that the sum of a sufficiently large number of such waves with random relative phases and the correct angular divergence will produce a Gaussian random field with the appropriate speckle characteristics for a beam from a DPP.^{72} The propagation angles of the plane waves must be distributed over the solid angle of the focusing lens, so their total spread is $1/f$, where *f* is the f-number of the lens. The resulting interference of plane waves creates laser speckles with minimum width $f\lambda 0$ and length $8f2\lambda 0$. For the laser configuration used in many contemporary ICF experiments and in the glass-laser simulation results presented in this paper (i.e., f/6.7 lenses and a 0.351-*μ*m laser wavelength), this results in a characteristic speckle width and length in each beam of approximately 2 and 126 *μ*m, respectively. Since the effects of SSD or ISI are not considered here, the laser speckles in our simulations are essentially fixed (in space) and modulated in time by the varying amplitude of the laser intensity. Finite bandwidth is modeled by introducing a spectrum of closely spaced, discrete frequency components in each of the plane waves used for generating laser speckles.

To estimate thresholds for the absolute TPD and SRS instabilities, we adopt an iterative numerical technique developed by Follett *et al.*^{31} The first step in this procedure is to establish a practical criterion for the existence of an absolute instability in an LPSE simulation with a finite run time. For our purposes, we consider a simulation to be absolutely unstable if the maximum electron-density perturbation reaches a value that is at least 10 orders of magnitude larger than the initial noise seed after a run duration not less than 100 ps. With this criterion, laser intensity thresholds for absolute TPD and SRS can be estimated by first making an initial guess for the threshold value and then running the LPSE code to see whether the simulation is stable or unstable. Depending on the result, the laser intensity is then adjusted upward or downward by a set increment and the code is run again. This procedure is repeated as many times as necessary in order to bracket the threshold value. Once upper and lower bounds have been established, the uncertainty in the threshold estimate can be reduced by repeatedly halving the bounding interval and subsequently running the LPSE code with updated laser intensities. In this study, we used an initial laser-intensity increment that was set to 1/3 of the initial guess, which implies that once upper and lower bounds are determined, the associated uncertainty is reduced to 1/6 of this value. Repeating this procedure five additional times then lowers the final threshold uncertainty to $(1/6)(1/2)5<$ 1%. Additional details on this iterative procedure can be found in Ref. 31.

We now turn to a presentation of selected results obtained with the LPSE code that model CBET and the absolute forms of the TPD and SRS instabilities in the context of direct-drive ICF. Results are presented for two different classes of laser drivers: Nd:glass lasers and ArF lasers; these appear in Secs. IV and V, respectively.

## IV. RESULTS FOR GLASS LASERS

In this section, we discuss two- and three-dimensional LPSE simulations of CBET, TPD, and SRS driven by frequency-tripled Nd:glass laser light. As we shall see, such lasers require a Gaussian bandwidth of about 8.5 THz ( $\Delta \omega /\omega 0\u22431%$) to restore absorption losses from CBET scattering in an OMEGA-scale ICF target and a somewhat larger bandwidth—approximately 13 THz ( $\Delta \omega /\omega 0\u22431.5%$)—to mitigate the absolute forms of TPD and SRS on an ignition-scale platform. In modeling these results, it is assumed that some sort of nonlinear optical technique has been applied at the end of the propagation chain to broaden the emission spectra from this type of laser. Promising methods of bandwidth broadening include parametric amplification via the FLUX laser system, as described in Ref. 25. A second possible approach for realizing broad laser bandwidth with glass lasers is to exploit the process of SRRS,^{26} which is a subject that we discuss more fully in Sec. V.

### A. CBET scattering in an ICF target

Let us first consider a series of LPSE simulations that model CBET in two spatial dimensions under plasma conditions similar to those found on the OMEGA laser. Figure 2(a) depicts the set-up for these simulations in which 16 symmetrically arranged, equal-intensity beams from frequency-tripled Nd:glass lasers irradiate the plasma atmosphere surrounding a solid, 720-*μ*m-diameter CH target. The CH plasma has temperatures $Te=2$ keV and $Ti=1\u2009keV$, and is modeled in our simulations as a single ion species with an average charge Z = 3.1, an atomic mass A = 5.3 (ion mass in amu) and an ion-to-electron mass ratio $mi/me=10\u2009319$. (This assumes that CH is comprised of 42% carbon and 58% hydrogen.) The electron-number-density and Mach-number profiles for these simulations are derived from the plot in Fig. 2(b), which shows results from a one-dimensional simulation of an OMEGA pellet implosion performed with the radiation-hydrodynamics code LILAC.^{73} Note that the laser configuration depicted in Fig. 2(a) is similar to one considered in a previous CBET study that modeled a smaller diameter target with analytical plasma-density and flow-velocity profiles.^{74} Figure 3 displays two LPSE simulation results for the configuration sketched in Fig. 2 using monochromatic, *s*-polarized laser light. Both simulations model a 1600 × 1600 *μ*m^{2} spatial region that is discretized on a computational grid using 40 320 × 40 320 zones (∼10 cells/wavelength). Results show the magnitude of the steady-state electric field for the 16 beams after approximately 10 ps, once a steady-state had been established. Each beam in these simulations had an average intensity of $1.5\xd71014\u2009\u2009W/cm2$ and included the effects of laser speckles and DPPs. The average intensity here is defined as the peak laser intensity that a plane wave beam would have for the same width and flux. The speckle patterns in the beams correspond to f/6.7 lenses and were generated by launching 50–100 beamlets from each laser source with fourth-order super-Gaussian distributions. The width of each beam was also chosen to be 424 *μ*m, which is about 18% wider than the radius of the target. Figure 3(a) presents a case in which ion acoustic waves were disabled in the LPSE code so that CBET scattering could not occur. In that case, approximately 93% of the light was absorbed in the coronal plasma. The plot in Fig. 3(b) modeled the same laser and plasma conditions as in Fig. 3(a), but in this case the effects of ion acoustic waves were included in the LPSE simulation, with the result that laser absorption was reduced to 41%. We also see from Fig. 3(b) that the inclusion of CBET caused some of the laser light to be scattered away from the highly absorptive region near the critical surface, which cast a visible shadow (penumbra) in the irradiation profile. Note that in Fig. 3(b), ion-acoustic waves were subject to ion Landau damping at a constant rate of $\nu a=0.1kacs$, where *k _{a}* is the wavevector of ion acoustic waves. For these simulations, ion acoustic waves were computed using a second-order finite difference algorithm, while the laser light was evolved in time using the spectral solver in LPSE.

We next consider the effect of finite bandwidth on laser absorption in the presence of CBET scattering. Laser bandwidth is included in our LPSE simulations by using a series of 50–100 closely spaced spectral lines arranged symmetrically on either side of a central laser frequency. The intensities of the lines follow a Gaussian distribution whose full-width-at-half-maximum corresponds to the laser bandwidth $\Delta \nu $. The maximum spacing between these lines is sufficiently small to expect an equivalence with the results from a continuous laser spectrum.^{78} Additionally, the different frequencies are randomly phased and three-run ensembles are performed to compute an average absorption in the coronal plasma. The plot in Fig. 4 shows the results of including this bandwidth model in our simulations. We see that at a fractional bandwidth of about 8.5 THz ( $\Delta \omega /\omega 0\u22431%$ for 351-nm laser light), absorption losses due to CBET scattering are nearly eliminated. This result is consistent with findings from previous studies of CBET that modeled different plasma conditions and polarization-smoothed laser light. The point labeled “Ω” in Fig. 4 denotes the current upper bound of the bandwidth for the OMEGA laser, which is not large enough to appreciably mitigate CBET. It may be possible, however, to achieve significantly larger bandwidth values and restore absorption losses using FLUX^{25} or the SRRS technique.^{26} Note that the laser light was modeled as *s*-polarized in these simulations in order to maximize the occurrence of CBET for the purposes of this study. We should also point out that successful CBET mitigation will likely increase the laser intensity at the quarter-critical density, which could lead to enhanced hot-electron production if the laser bandwidth is insufficiently large to also defeat absolute TPD and SRS.

### B. Thresholds for absolute TPD and SRS

Let us now examine the impact of finite bandwidth on the absolute TPD and SRS instabilities using frequency-tripled, Nd:glass laser light. We consider the laser configuration shown in Fig. 5, which consists of 6 overlapped, equally intense f/6.7 beams that form the edges of a right hexagonal pyramid with each beam incident at an angle of 23° relative to the central axis. This arrangement is similar to the geometry of a single cone of beams on the OMEGA laser. Using the LPSE code, we model a three-dimensional volume of plasma near the quarter-critical surface for two different sets of conditions. The first set of simulations model a plasma with electron and ion temperatures of 2 keV and 1 keV, respectively, and a linear density profile that ranges in the laser direction from $ne/nc=0.22$–0.26 on a 32 × 10 × 10 $\mu m3$ numerical grid with 480 × 150 × 150 computational cells. This corresponds to a density gradient scale length of $Ln=|\u2202\u2009log\u2009ne/\u2202x|\u22121$ = 200 *μ*m and a collisional (amplitude) damping rate of 1.3 ps^{−1}. The second set of simulations model a plasma with electron and ion temperatures of 4 keV and 2 keV, respectively, for the same density range over a 96 × 10 × 10 $\mu m3$ grid (using 1440 × 150 × 150 cells). This corresponds to *L _{n}* = 600

*μ*m, which is close to the value one might expect to find on an ignition-scale ICF platform. The collisional damping rate used in this case was 0.5 ps

^{−1}. Landau damping was also modeled in all the simulations, but was not large enough to appreciably affect the results. This conclusion follows from the fact that for both TPD and SRS, the decay mode with the lowest absolute threshold has one daughter wave with a wavevector near zero and one daughter electron plasma wave with a wavevector that is nearly equal to that of the drive laser.

^{39}Consequently, the quantity $ke\lambda D$ can be approximated as $k0\u2009[kB\u2009Te/(4\u2009\pi \u2009e2\u2009ne)]1/2$, where

*k*is the wavenumber of the electron plasma wave and

_{e}*λ*is the Debye length.

_{D}^{75}For the two sets of plasma conditions modeled in our simulations, we find $ke\lambda D\u22430.11$ and 0.15, respectively—each of which is substantially below the threshold value of 0.3 where the effects of Landau damping are significant.

^{76}Note also that for the absolute TPD and SRS-backscatter simulations performed in this study, the solutions were enveloped at a frequency corresponding to $n0/nc$ = 0.25.

Absolute TPD and SRS thresholds for the two sets of plasma conditions described above are shown in Fig. 6 for different values of laser bandwidth. Each of the data points shown in this figure is the average of a three-run ensemble of LPSE simulations with different realizations of speckle pattern, polarization smoothing, and phase, where error bars denote one standard deviation from the mean value. We see from Fig. 6 that for *L _{n}* = $600\u2009\mu m$, absolute SRS backscatter has a lower threshold than absolute TPD, which is consistent with SRS being the dominant source of hot electrons in longer scale length plasmas, such as those found on the NIF.

^{50}Additionally, while it is generally true for a single plane wave beam that SRS sidescatter has a lower absolute threshold than backscatter, that is not the case here. That is because SRS sidescatter also has weaker multi-beam coupling for ICF-relevant conditions.

^{39}Note that the density range used to model absolute SRS sidescatter in our simulations was $ne/nc$ = 0.18–0.2, with an envelope frequency corresponding to $n0/nc$ = 0.2.

From a target design point of view, there are at least two ways to interpret the results presented in Fig. 6. The first is that for a given density gradient scale length and bandwidth, the instability with the lowest absolute threshold serves as an approximate upper bound for the laser intensity that can be used to implode an ICF target. A second possible interpretation is that for a fixed laser intensity, one can prescribe the laser bandwidth required to avoid absolute instabilities. As an illustration of the latter, consider current, ignition-scale target designs using 351-nm laser light, which have $Ln\u2243600\u2009\mu m$ and an overlapped, incident intensity in vacuum of approximately $1015\u2009\u2009W/cm2$. Near the quarter-critical-density surface where TPD and SRS occur, this intensity is reduced due to collisional absorption, and potentially a small degree ( $\u223c10%$) of CBET scattering as well, if the bandwidth of the laser light is insufficiently broad to suppress it. Taking these factors into consideration, a reasonable estimate for the quarter-critical intensity is then $35%\u221260%$ of the incident value, which is indicated by the two dashed horizontal lines in Fig. 6.

Inspection of Fig. 6 shows that a laser bandwidth of approximately 13 THz (equivalent to a fractional bandwidth $\Delta \omega /\omega 0\u2243$ 1.5%) would be sufficient to raise all the absolute instability thresholds above the ignition-scale design range. Note that Follett *et al.*^{39} reached a similar conclusion based on scaling arguments and LPSE simulation results for a smaller value of *L _{n}*. That work also examined other considerations such as beam geometry and angular dispersion and found that absolute TPD and SRS thresholds were only weakly sensitive to variations of such factors. We should emphasize here that convective SRS has not been addressed in this study and that additional work is required to assess the effect of bandwidth on that instability in multi-beam ICF configurations. Even if convective SRS is not an important factor in ICF target designs, using broadband lasers to mitigate the absolute forms of the TPD and SRS instabilities will lead to an enhanced drive intensity, which could exacerbate the deleterious effects of convective SRS.

## V. BROADBAND LASER SOURCES

In Sec. IV, we presented LPSE simulation results showing that a laser bandwidth of about 13 THz is sufficient to mitigate most of the parametric instabilities on an ignition-scale platform using 351-nm laser light. Although the Nd:glass lasers used for ICF research today have bandwidths far below this value, the spectra from such lasers could likely be broadened to multi-terahertz levels by exploiting a nonlinear optical process such as SRRS in a gaseous diatomic medium. Another approach to achieving the broad bandwidths required for parametric instability suppression is to adopt the ArF excimer laser as an ICF driver. This would enable higher and more-symmetric ablation pressures in future, direct-drive target designs. In this section, we discuss these two approaches in greater detail.

### A. Stimulated rotational Raman scattering

The SRRS process occurs when a laser excites rotational quantum states of a diatomic molecule along its path of propagation. The rapid de-excitation of the molecule results in an emitted light spectrum that contains, in general, both lower (Stokes) and higher (anti-Stokes) frequencies. Historically, SRRS has been regarded as a pernicious phenomenon in ICF because it can significantly reduce beam uniformity at the focal plane of the target. In fact, many large-scale laser systems such as the NIF are designed to mitigate SRRS by propagating beams through enclosures filled with a monatomic gas (e.g., argon) before the light enters the target chamber.^{77} Despite the perception of SRRS as being deleterious for ICF, though, it can, under the right conditions, also provide a practical mechanism for enhancing the bandwidth of ICF drivers in order to mitigate parametric instabilities.

Previous experimental studies have demonstrated that the SRRS process in atmospheric nitrogen can generate a discrete multiline spectrum with a line separation of approximately 0.5 THz that stretches across a total frequency range of several percent of the laser frequency.^{78} It is believed that the effect of such a spectrum on CBET, TPD, and SRS would be similar to that for a continuous broadband (Gaussian) laser because the line separation is less than the growth rates of those instabilities under typical ICF conditions.^{78} (Note that a similar discrete form of bandwidth is also envisioned in the StarDriver® concept in which numerous low-energy, small-aperture, monochromatic beams operating at slightly different discrete wavelengths are combined to form a multiline spectrum spanning a wide swath of frequency space.^{79}) The amount of SRRS that occurs, and the rate at which it develops, though, depends on several key factors, including the laser intensity, the interaction length, and the medium through which the light propagates. Additionally, SRRS can be self-seeded if the frequency spectrum of the laser is sufficiently broad or it can be seeded by a second, co-propagating beam of lesser power.^{26} Recent experiments on a single 40-J beam of the Nike laser at NRL, for example, have demonstrated that self-seeded SRRS in air can increase laser bandwidth from ∼1 to 5 THz at 248 nm with only a modest degradation of the focal distribution.^{80} Detailed kinetic simulations performed at NRL indicate that even larger enhancements are possible and that SRRS can be applied at greater laser energies. Another appealing aspect of SRRS is that it is compatible with all laser wavelengths used in connection with ICF research (i.e., 527 nm, 351 nm, 248 nm, and 193 nm). For glass lasers in particular, SRRS offers a promising approach for realizing the multi-terahertz bandwidths required for parametric instability suppression.

### B. The ArF laser driver

Due to its deep ultraviolet wavelength, the ArF laser has long been recognized as an attractive driver for ICF. In the 1970s and 1980s, researchers in the United States^{81} and Japan^{82} constructed two 100-J class ArF systems, but work in this area was ultimately abandoned in the belief that frequency-tripled Nd:glass lasers would provide sufficient energy coupling for driving ICF implosions. Over the last 40 years, though, the identification of parametric instabilities as a principal obstacle to success in laser fusion, as well as a need to develop effective strategies for their mitigation, has led to a resurgence of interest in broadband lasers, particularly ArF. In addition to having the shortest wavelength (193 nm) that can scale to the high energy and power required for high fusion yield, the ArF laser also possesses a broad native bandwidth of approximately 10 THz.^{27–29} As we shall see later in this section, these native properties of ArF laser light are highly effective at suppressing CBET, as well as the absolute forms of TPD and SRS. Additionally, the ArF laser has a higher target-coupling efficiency than a frequency-tripled, Nd:glass laser, as demonstrated in a recent study.^{83,84}

Although there are certain technological challenges associated with building a high-energy/high-power ArF-laser system presently (most notably, the development of large-aperture optical components with fluoride-resistant coatings and high damage thresholds at 193 nm), no obvious “showstoppers” have been identified that would preclude the use of an ArF laser as a future ICF driver.^{29} Assuming such challenges can be met, the ArF laser is a potential game changer in the field of ICF because its deep-ultraviolet wavelength and broad bandwidth would permit the use of significantly larger laser intensities and ablation pressures than can be utilized currently with frequency-tripled Nd:glass lasers. This, in turn, would facilitate the implosion of more efficient, lower-aspect-ratio targets that are more resistant to hydrodynamic instabilities and require less driver energy and illumination uniformity to achieve high yield. Furthermore, an ArF driver could exploit many of the advantageous technologies previously demonstrated with KrF excimer lasers. These include a versatile ISI beam-smoothing capability,^{18} as well as laser zooming,^{85} which can further increase the energy coupled to an imploding target. Let us now consider the results of several numerical simulations that illustrate the advantages of the ArF laser for direct-drive ICF.

We begin by examining the efficacy of ArF laser light for mitigating CBET in the target configuration sketched in Fig. 2. As in the previous example using 351-nm laser light, the ArF simulations model 16 *s*-polarized lasers arranged symmetrically around an OMEGA-scale, solid-shell CH target where the total width of each beam is about 18% wider than the target's diameter. Once again, the f/6.7 beams include the effects of laser speckles and DPPs, but in this case, the electron temperature, ion temperature, and average single-beam laser intensity have the values 4.8 keV, 1.4 keV, and $5\xd71014\u2009W/cm2$, respectively. Additionally, the density and velocity profiles used in this set of simulations were taken from one-dimensional FastRad3D radiation-hydrodynamic simulations^{86} of a conventional hot-spot implosion driven by an ArF laser. Inspection of Fig. 7 shows that for a monochromatic ArF laser in the absence of CBET, about 96% of the light is absorbed in the coronal plasma. When CBET is included in the simulations, though, the absorption drops to 61%. We also see, however, that at the 10-THz native bandwidth of an ArF laser, nearly all of the absorption losses caused by CBET scattering are recovered.

*T*= 2.6 keV,

_{e}*T*=1 keV, and $Ln=211\u2009\mu m$. For each color, note that there are two sets of LPSE simulation results shown: the instability thresholds for six, overlapped polarization-smoothed beams in three dimensions (which were computed using collisional amplitude damping rates of 0.9 and 2.7 ps

_{i}^{−1}for frequency-tripled Nd:glass and ArF laser light, respectively) and the thresholds for a single, two-dimensional,

*p*-polarized beam (for which the effects of collisional absorption were neglected). Each data point in Fig. 8 represents the average of a three-run ensemble of LPSE runs to account for variations in the random phase, speckle pattern, and polarization smoothing of each beam; error bars denote one standard deviation from the mean value. The two horizontal dotted lines in this plot are the results of a linearized theory for a single monochromatic plane wave beam, which predicts a TPD threshold given by

^{45}

*T*= 4 keV,

_{e}*T*= 2 keV, and $Ln=400\u2009\u2009\mu m$, which characterize the conditions that one might expect to have in a direct-drive experiment on the NIF. As before, each data point in Fig. 9 is an average of a three-run ensemble to account for variations in the random phase, speckle pattern, and polarization smoothing of each beam, where error bars denote one standard deviation from the mean value. Additionally, each of the LPSE simulations was run for at least 150 ps and included the effects of Landau damping as well as collisional damping (with amplitude damping rates of 0.5 and 1.5 $ps\u22121$ for frequency-tripled Nd:glass and ArF lasers, respectively). The two dotted lines in Fig. 9 denote the predictions from a linearized analytical theory for absolute SRS backscatter, which is given by the expression

_{i}^{46}

## VI. SUMMARY AND CONCLUSION

Ideally, the laser-plasma coupling in the corona of an ICF target is dominated by inverse bremsstrahlung (collisional) absorption.^{1} However, this is not necessarily what occurs in practice, as various forms of parametric instabilities can arise to complicate the laser absorption process.^{2} One such instability that is particularly troublesome is CBET, which is a form of stimulated Brillouin scattering that occurs when two or more beams cross each other in a sonic plasma, with the result that laser energy can be resonantly transferred from one beam to another.^{8,9} CBET can significantly reduce laser absorption in an ICF implosion and can also compromise the implosion symmetry by scattering light in unwanted directions. In addition to CBET, there are other high-frequency instabilities that play a significant role in direct-drive ICF, namely, the absolute TPD and SRS instabilities,^{2,3} which can also generate deleterious supra-thermal (>50 keV) electrons that penetrate the high-density thermonuclear fuel, raise its adiabat, and hence spoil high compression.^{4–6}

Using the LPSE code,^{30} the efficacy of broad laser bandwidth for suppressing CBET and the absolute forms of the TPD and SRS instabilities has been investigated in this paper. It was shown that for frequency-tripled Nd:glass lasers, a Gaussian bandwidth of about 8.5 THz ( $\Delta \omega /\omega 0\u22431%$) is required to restore absorption losses from CBET scattering in an OMEGA-scale target and a somewhat larger bandwidth—approximately 13 THz ( $\Delta \omega /\omega 0\u22431.5%$)—is needed to mitigate absolute TPD and SRS on an ignition-scale platform. With an ArF laser driver,^{29} which has a native bandwidth of approximately 10 THz and a wavelength of 193 nm, two-dimensional LPSE simulations modeling realistic ICF plasma conditions suggest that nearly all of the absorption losses caused by CBET scattering can be recovered. It was also demonstrated in this paper that the ArF laser can enable absolute TPD and SRS-backscatter threshold intensities above $1015\u2009\u2009W/cm2$ for typical, direct-drive, ICF plasma conditions, which is appreciably larger than thresholds corresponding to frequency-tripled, Nd:glass laser light, even with enhanced bandwidths up to 20 THz ( $\Delta \omega /\omega 0\u22432.3%$). This represents a significant advantage of ArF drivers in ICF and facilitates the use of higher laser intensities and ablation pressures in future, direct-drive target designs.

## ACKNOWLEDGMENTS

The authors wish to thank Dr. A. J. Schmitt and Dr. S. E. Bodner for useful discussions and Mr. K. Obenschain for assistance with computing facilities. This work was made possible through a computing grant from the High-Performance-Computing Modernization Project within the U.S. Department of Defense, but was performed under the aegis of the U.S. Department of Energy's National Nuclear Security Administration (NNSA), Advanced Research Projects Agency-Energy (ARPA-E), and Fusion Energy Sciences (FES) programs.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jason W. Bates:** Writing – original draft (lead); Writing – review & editing (lead). **Jude Kessler:** Investigation (supporting). **Russell K. Follett:** Methodology (equal). **John Shaw:** Software (lead). **Stephen Philip Obenschain:** Project administration (equal). **Jason Frank Myatt:** Software (supporting). **James L. Weaver:** Investigation (supporting). **Matthew F. Wolford:** Investigation (supporting). **David Martin Kehne:** Investigation (supporting). **Matthew Myers:** Investigation (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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